Optimal Transport for Inverse Problems
summary
Using Wasserstein loss instead of L2 for PDE and ML inverse problems aligns transport features of model and data so gradient-based optimization avoids many spurious stationary points that trap L2, notably in hyperbolic (wave/velocity) inversion. Empirical results show residuals can drop near zero where L2 stalls; stability improves because the metric behaves like a weighted H^{-1} norm, filtering high-frequency noise and providing implicit regularization. Demonstrated on seismic/source localization, deconvolution and imaging, with links to mean-field limits and modified gradient flows for large-scale, nonconvex parameter estimation.
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chapters (10)
- 0:00Intro
- 0:28Inverse Problems are Everywhere
- 1:36More General Inverse Problems
- 2:26The Role of the Wasserstein Metric
- 3:28Regarding Convexity
- 5:23Property One: Better Convexity
- 6:35The Connection Between W, and the Sobolev Norm
- 6:56The Implicit Regularization
- 7:29Property Two Better Stability
- 9:00Summary and Extensions
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